in terms of the Jacobi theta function \(\vartheta _1\)  and its first derivative w.r.t. \(z\) . As warned by Abramowitz and Stegun [1]}, there is a bewildering variety of notations for the theta functions. The one we use here, given by [1]} and [3]}, is such that they have the following Fourier representation:
\(\vartheta _1 (z, ^{-\chi })= 2\sum _{n=0}^ {\infty } (-1)^n ^{-\chi (n+1/2)^2} \sin \left[(2n+1 )z\right], \) 
